Forschungsinstitut zur Zukunft der ArbeitInstitute for the Study of Labor

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Earmarking and the Political Support of Fat Taxes

IZA DP No. 9335

September 2015

Helmuth CremerCatarina GoulãoKerstin Roeder

Earmarking and the Political Support of Fat Taxes

Helmuth Cremer Toulouse School of Economics (GREMAQ, IDEI, IUF)

and IZA

Catarina Goulão

Toulouse School of Economics (GREMAQ, INRA)

Kerstin Roeder University of Augsburg

Discussion Paper No. 9335 September 2015

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IZA Discussion Paper No. 9335 September 2015

ABSTRACT

Earmarking and the Political Support of Fat Taxes* A fat and a healthy good provide immediate gratification, and cause health costs or benefits in the long run, which are misperceived. Additionally, the fat good (healthy good) increases (decreases) health care costs by increasing (decreasing) the probability of suffering from a chronic disease in the future. Individuals differ in income and in their degree of misperceptions concerning the health effects of the consumption of fat and of healthy goods. The level of the fat tax is determined through majority voting. Individuals vote according to their misperceived utility function. Consequently, excessive fat consumption is not due to a self-control problem but due to information deficiencies or cognitive inability to process information. A fraction of the fat tax proceeds is “earmarked” to reduce health insurance premiums while the remaining fraction finances a subsidy on the healthy good. This earmarking rule is determined at a constitutional stage to maximize utilitarian or Rawlsian welfare, anticipating the induced political equilibrium. We show that the fat tax in the political equilibrium is always lower than the utilitarian fat tax. This is no longer necessarily true with a Rawlsian objective. The determination of the optimal earmarking rule is quite complex. Even in the utilitarian case, it is not just used to boost political support for the fat tax. Instead, it may involve a tradeoff between the fat tax and the healthy good subsidy. JEL Classification: I12, I18, D72 Keywords: obesity, fat tax, misperception, voting, earmarking Corresponding author: Helmuth Cremer Toulouse School of Economics 21 Allée de Brienne 31015 Toulouse France E-mail: helmuth.cremer@tse-fr.eu

* We thank Pierre Pestieau and all participants of the EHEW in Toulouse for their helpful comments and suggestions. Financial support from the Chaire “Marché des risques et creation de valeur” of the FdR/SCOR is gratefully acknowledged.

1 Introduction

Worldwide obesity has more than doubled since the 1980s. In 2014, 33% of the US

and 23% of the European populations were obese. More than 70% of the US and UK

population are overweight, and worldwide there are examples of similar patterns even

among developing countries such as Argentina and Chile, or middle eastern countries

such as Kuwait, Qatar, and Saudi Arabia.1 Because of such high prevalence rates,

obesity is considered an epidemic despite being non-infectious.

The statistics on the prevalence of obesity are worrying because, along with tobacco

consumption, overweight and obesity are risk factors of non-communicable chronic dis-

eases (NCDs), which are presently the leading cause of deaths worldwide. Examples of

NCDs are heart disease, stroke, diabetes, musculoskeletal disorders like osteoarthritis,

and some sorts of cancer. According to the WHO, NCDs account for over 60% of the

total deaths in the world, and half of these deaths occurred in low- and middle-income

countries before age 70.2

In addition to being a main factor in premature mortality, NCDs are also responsible

for extended periods of morbidity and dependency. These impediments may dramatic-

ally reduce individuals’ quality of life. Additionally, they lead to a significant increase in

present health care costs worldwide even if an individual’s full lifetime is accounted for.

For example, Cawley and Meyerhoefer (2012) estimate obesity medical costs of 2,741

USD, which translates in 20% of the US health expenditure. Finkelstein et al. (2008)

find that the severely obese white women have the highest full lifetime health care cost

of 29,460 USD.3

There are many factors put forward as reasons for the obesity spread (Cutler et al.,

2003), but in the end it typically boils down to unbalanced caloric intakes compared

to energy needs. Consequently, a policy intervention aiming at eradicating, or at least

slowing down the epidemics has to affect diet and physical activity choices. Of course

obesity also has a genetic component. However, genetics cannot explain the dramatic

increase in obesity rates over the last three decades (Hill and Peters, 1998).

In this paper we analyze the political support of a “fat tax”, imposed on fattening

1Overweight and obesity prevalence rates statistics available from the World Health Organization

(WHO), : 2461? = accessed on June 26, 2015.2http://www.who.int/mediacentre/factsheets/fs355/en/, accessed on June 26, 2015.3See as well Finkelstein et al., 2003; Finkelstein et al., 2011; Finkelstein and Yang, 2011; Thorpe et

al., 2004; Wang et al., 2008; and Wang et al., 2011; among others.

1

food and beverages, combined with a subsidy on healthy goods. These are amongst

the policy instruments with the potential to slow down the obesity epidemics. We also

examine how the government can affect this political support by “earmarking” the tax

revenue.

In reality fat taxes exist, but remain relatively rare and their level is not very sig-

nificant. A fat tax was first introduced in Denmark in October 2011, yet withdrawn

one year later. Currently, fat taxes are being discussed in several countries such as the

UK or France.4 In 2009, 33 states in the USA taxed soft drinks, and more recently

France, Norway and Mexico have imposed similar taxes (Brownell et al., 2009). The re-

lative scarcity of fat taxes in practice suggests that such a policy may not be sufficiently

attractive from a political perspective.

Intuitively, one can expect the following factors to affect the political support of fat

taxes. First, they are likely to be regressive. Low-income individuals consume more

fattening food and beverages, in particular, because they often misperceive the effects

of diet and physical activity in the short and the long run (see Vereecken and Maes,

2010; and Bauman et al., 2012; among others).5 Consequently, the median income

individual bears a larger than average burden of the tax; this may lead to a lower than

optimal level of the fat tax in a voting equilibrium. Second, irrespective of distributional

considerations, misperception in itself may imply that individuals underestimate the

benefits of the tax which undermines its political support. Third, in countries with

SHI (Social Health Insurance) or NHS (National Health System), individuals may not

properly account for the social health care costs caused by their unhealthy lifestyle.

In this paper we study the political determination of fat taxes in a simple setting. We

show that the equilibrium is typically not optimal, neither from a utilitarian nor from a

Rawlsian perspective. More precisely, the equilibrium level of the fat tax is always lower

than the utilitarian optimum, but it may be lower or higher than the Rawlsian solution.

However, this is just a prelude to the main question we ask, namely if and how the

government can affect the political support for fat taxes through “an accompanying”

policy of earmarking to bring the voting solution closer to the optimum.

4See “UK could introduce a fat tax, says David Cameron", theguardian.com, Tuesday 4 October 2011

13.50 BST, : 201104−−−−,

accessed on June 26, 2014 and IGF (2008); Bonnet et al. (2009); and Allais et al. (2010).5However the regressive nature of a fat tax may be compensated by the health benefits it induces

through the improvement of diet.

2

We consider an economy wherein agents differ in their income and in their degree of

misperceptions concerning the health effects of fat and of healthy goods. Consumption

of the fat good gives rise to immediate satisfaction but also causes adverse health effects,

which are misperceived. In turn, the consumption of the healthy good has beneficial

health effects, which are misperceived as well.6

The fat good consumption increases the probability of suffering from NCDs in the

future. We assume that all sick individuals are treated within a national health system,

which is financed by a uniform capitation fee. Therefore, individuals’ fat consumption

imposes higher taxes on all due to the need to balance the budget of the national health

system. In such a setup, we investigate the role of a fat tax aiming to correct: (i)

individuals’ misperceptions concerning health effects, and (ii) the negative effect of an

increase in the health care costs. Since a fat tax constitutes a new source of financing

the capitation fee can be cut down. Additionally, the capitation fee can be reduced

further because health care expenses decrease with a decrease in fat consumption.

We consider the possibility that the fat tax proceeds are used either to contribute to

the financing of the health care system or to subsidize the consumption of some healthy

good or activity (see Cremer et al., 2010; and Goulão and Pérez-Barahona, 2014). In

other words, the proceeds of the fat tax are earmarked so that some fraction is used

to finance health care while the remaining part finances a subsidy on the healthy good

consumption (for the later see Yaniv et al., 2009).7 This earmarking rule is determined

by a welfare maximizing (utilitarian or Rawlsian) government at a constitutional stage

which precedes the vote over the fat tax. It is by now well known that earmarking is at

best useless in a purely normative setting. However, it has interesting effects when all

or part of the policy instruments are determined by a political process. In particular,

we show how earmarking of the fat tax revenues can be used to boost the political

support of a more welfare enhancing fat tax. The voting equilibrium tax rate increases

as the fraction of tax revenues devoted to the financing of health insurance increases.

6There seems to exist a consensus that the population is generally poorly informed about the attrib-

utes of a healthy diet (Vereecken and Maes, 2010; Grimes et al., 2009; and Schwartz et Byrd-Bredbenner,

2005) and there is evidence that greater nutritional knowledge is related to better nutrition (Pollard et

al., 2010; and Vereecken and Maes, 2010).7This is pretty much in line with the allocation of fat tax suggested by Leicester and Windmeijer

(2004, page 10) who argue that “ Funds from a ‘fat tax’ could be used to finance healthy eating advice

in schools or on television, for the NHS, etc. Alternatively, the income could be used to reduce other

taxes or provide subsidies on healthy foods or exercise equipment.”

3

Consequently, one might expect that a utilitarian government would choose to devote

a larger share of tax revenue to health expenses than otherwise optimal. We show that

this is not necessarily true because an increase in this share tends to decrease the subsidy

on the healthy good (which is also too low at the utilitarian solution). Similar effects

are at work in the Rawlsian case with the added complexity that the voting equilibrium

tax level may then actually exceed the optimal level.

Our paper contributes to a growing literature on sin taxes. O’Donoghue and Rabin

(2003; 2006) study precisely the relevance of sin taxes in view of correcting individuals

misperceptions. They conclude that they may even be Pareto improving since they help

people with self-control problems without imposing losses to fully rational people.

Some political economy aspects of sin taxes have been studied by Haavio and Ko-

takorpi (2011). They show how voting over a sin tax can work as a commitment device

for individuals with self-control problems. Our approach departs from theirs in two

main directions. First, we consider that individuals vote according to their misper-

ceived utility function. In other words, we assume that excessive fat consumption is not

due to a self-control problem. Instead, an individual’s misperception of health effects is

due to information deficiencies or cognitive inability to process information. Moreover,

we suppose that higher income individuals have lower misperception levels. Second, we

consider the impact of fat consumption on the costs of the health care system. This

feature together with the possibility of earmarking the funds of the fat tax to subsidize

a healthy activity make the political problem more complex but generates interesting

equilibria.

The remainder of the paper is organized as follows. Section 2 describes our economic

environment. Then, from a normative perspective, the optimal fat tax and budgetary

rule are analyzed in Section 3. The political fat tax is determined in Section 4. The

constitutional stage is analyzed in Section 5 and Section 6 provides some concluding

remarks.

2 The economic environment

The economy consists of individuals who differ in income which is distributed on the

support [ ] ∈ R+ according to the cumulative distribution function (). Popula-

tion size is normalized at one. Individual’s utility depends on the consumption of an

4

unhealthy, high fat or sugar good , (e.g., potato chips or sodas), a healthy good

(e.g., vegetables), and a numeraire good (e.g., leisure). To simplify the exposition we

will often refer to the unhealthy commodity as “fat good”, but our analysis applies just

as well to high sugar products. All producers’ prices are normalized at one. Fat goods

are taxed at a rate ≥ 0 and healthy goods are subsidized at a rate ≥ 0.The satisfaction (and health benefits) individuals obtain from the fat good is reflected

by () with 0() 0 and 00() 0. In the long run, however, higher consumption

of fat causes overweigh or obesity and comes with health problems. These negative

effects of fat good consumption are captured by the harm function () with 0() 0

and 00() 0. In the short run, individuals may ignore part of these negative long-run

effects. Their perceived harm function is given by () where ∈ [0 1]. We assumethat ≡ () decreases with income, i.e.,

0() 0. The empirical support for this

correlation is broad; see, e.g., Devaux et al., (2011), and McLaren (2011) for evidence

on the negative association of socio-economic status/education and obesity/nutritional

knowledge.

The healthy good provides a utility of () with 0() 0 and 00() 0. In

line with the empirical evidence these benefits are also misperceived by some agents

implying (), where ∈ [0 1]. Once again the degree of misperception depends onincome, ≡ (), and is higher the lower individuals’ income: 0() 0. Finally,

utility is linear in the numeraire good.

Consumption of fat and healthy goods determines the probability that an individual

will develop a chronic disease, ( ). This probability is increasing in the fat good

consumption and decreasing in the healthy good consumption. For simplicity we assume

2 = 0. The government supplies “free” health care to those suffering from

a chronic disease. The health care scheme provides universal and full coverage and is

financed by capitation fees . The cost of treating patients is normalized to one.

In sum, individual ’s perceived utility, b, is given by

b = + () + ()− () (1)

while the individual’s true utility, , amounts to

= + () + ()− () (2)

Since all patients are treated and cured ( ) does not directly appear in individuals’

5

utilities, but it of course affects utility indirectly through the taxes levied to finance

health care expenditure.

2.1 Individual optimization problem

Individuals choose their consumption of the fat and the healthy goods by maximizing

perceived utility specified by equation (1), subject to their budget constraint, i.e.

max

b = + () + ()− ()

s.t. = − − (1 + ) − (1− ) (3)

The first order conditions (FOCs) are given by

b

=− (1− ) +

0() = 0 (4)

b

=− (1 + ) + 0()−

0() = 0 (5)

These equations determine the demand functions ∗ ≡ ( ) and ∗ ≡ ( ).

The comparative statics with respect to the tax and subsidy, and with respect to the

misperception parameters are as follows:

∗

=− 1

00() 0 (6)

∗

=− 0()00()

0 (7)

∗

=1

00()− 00()

0 (8)

∗

=− −0()00()−

00() 0 (9)

Not surprisingly, a higher subsidy on healthy goods and a lower misperception of their

health benefits increase their consumption. Similarly, a higher tax on fat goods and

a lower misperception of their negative health effects decrease an individual’s fat good

consumption. Since low-income individuals suffer from a higher misperception they tend

to consume more fat and less healthy goods than individuals with higher income and

lower misperception.

2.2 Equilibrium

In an equilibrium, public budgets need to be balanced. Health care costs are given byZ

[( ()) ( ())]d () ≡ ( ) (10)

6

They are financed by capitation fees and the revenue raised from the tax on the fat

good. The fat tax revenue is given byZ

( ())d () ≡ ()

A share ≤ 1 of fat tax revenue is earmarked to finance the health care system while

the remaining share 1 − is used to subsidize the healthy good consumption. The

government balances both the health care budget and the budget for the healthy good

subsidies. These two constraints determine and as functions of and . Specifically,

we have

(1− )() = () ⇒ ( ) (11)

+ () = ( ( )) ⇒ ( ) (12)

where () ≡ R

( ())d () is the average consumption level of the healthy good.

Note that we do not restrict to be positive. To understand this, observe that

equation (11) implies

1− =()

() (13)

In words, (1− ) measures the ratio between the budgetary cost of the subsidy on the

healthy good and the tax revenue collected on the fat good. With ≥ 0 and ≥ 0, wehave 1− 0 so that 1 is not possible. However, there is no good reason to rule

out the case where () () so that 0. In this case, the remaining part of the

subsidy on the healthy good is financed by the capitation fee, as shown by (12).

Assuming a positive fat tax, the effect of on ( ) and ( ) is given by

=(1− )(1− )

(1 + ) 0 (14)

= − (1− ) 0 (15)

where ≡ −() and ≡ () are the elasticities of society’s fatand healthy good consumption with respect to the fat tax and the subsidy respectively

and ≡ +()() 0. Note that a higher taxation of fat goods reduces

the average probability of chronic disease and thus health care costs both directly by

reducing and indirectly by increasing . In accordance with the empirical estimations

of Allais et al.(2010); Beatty and La France (2005); Harding and Lovenheim (2013); and

Huang and Lin (2000) among others, we assume that ≤ 1. Consequently, equation

7

(14) is positive and equation (15) is negative. In words, capitation fees decrease with the

fat tax rate, whereas the subsidy on the healthy good increases with it. The intuition

behind it is straightforward. If a 1% increase in the fat tax leads to a decrease in the

aggregate fat good consumption by less than 1%, a positive revenue from taxation is

generated. This revenue can be used to reduce capitation fees and increase the subsidy

on healthy activities.

The comparative static effects with respect to are given by

=

−(1 + )

0 (16)

= − ≶ 0 (17)

where ≡ ()() 0. The more of fat tax revenue is devoted to the healthcare scheme the lower are the subsidies on healthy goods. However, the impact of on

is ambiguous. The direct effect, measured by the second term in (17), is of course

negative, because a larger share of the fat tax proceeds is devoted to the financing of the

health insurance scheme. But there is also an indirect effect which goes in the opposite

direction. This is because the decrease in subsidies on the healthy good will tend to

increase health care cost.

For future reference, it is useful to express type- agents’ perceived and true utility

level by their indirect utility functions. Substituting ( ) and ( ), as defined by

equations (11) and (12), into the perceived and true utility functions, (1) and (2), yields

b( ) = − ( )− (1 + )()− (1− ( ))(( )) + ((( )))

+ (())− (()) (18)

( ) = − ( )− (1 + )()− (1− ( ))(( )) + ((( )))

+ (())− (()) (19)

Before analyzing the voting on the fat tax, we determine the socially optimal fat tax

and budgetary rule, which provide a benchmark against which the properties of the

political outcome can be assessed.

3 Normative benchmark

Since the public health care system is a prime ingredient of our setup, a “pure” laissez-

faire (LF ) solution is ruled out by definition. The situation which comes closest to a

8

conventional LF corresponds to the case where = 0 which (through the earmarking

rule) also implies = 0. It is plain that this solution is not efficient. To be more precise

there are two sources of inefficiency. The first one is the individuals’ misperception

of the long-term effects of the fat and healthy good consumption. In addition, with a

continuum of individuals health care costs are formally like an “atmosphere externality”.

Any given individual neglects the impact of his fat and healthy consumption on .

Interestingly, this externality occurs even when aggregate health damages are correctly

perceived by individuals. In other words, even when individuals are aware of equation

(10) their consumption decisions will neglect its impact on health care cost.

We also know from standard textbook theory that in a world of perfect information

these inefficiencies could be addressed through a system of individualized Pigouvian

taxes (on ) and subsidies (on ). We assume that such individualized taxes are not

feasible, for instance because of asymmetries of information. Instead, as shown in the

previous section, we restrict the government’s instruments to linear taxes and subsidies

with the same rate applied to all individuals.

Within this context we first examine the utilitarian solution. It is constrained by the

linear instruments ( ), but they are both under the control of the welfare maximizing

government. Note that with quasi-linear preferences, redistributive considerations do

not matter for the utilitarian government. Yet, a common concern with fat taxes is

their regressive nature; see, Allais et al. (2010); Frazão et al. (2007); Leicester and

Windmeijer (2004); and Powell and Chaloupka (2009); among others.8 To address the

redistributive nature of a fat tax we consider as well a Rawlsian government.

These solutions represent an interesting benchmark against which the political out-

come considered in the subsequent sections can be assessed. This is the reason why we

optimize with respect to and , the variable used in our political model, rather than

with respect to and . In a normative setting the change of variable is of no relevance.

Using and as decision variable would effectively be simpler and more intuitive. In

the utilitarian case this amounts to determining a Pigouvian tax and subsidy and the

is then determined purely mechanically from equation (13); see Appendix A.1.

8As noted by Katakorpi (2008), if low-income individuals have more elastic demand than high-income

individuals a fat tax may even be progressive.

9

3.1 Utilitarian welfare function

The utilitarian social planner chooses the fat tax and budgetary rule by maximizing

the sum over individual’s true net benefits of the fat and healthy good consumption as

given by equation (19)

max

( ) =

Z

( )d () (20)

The FOC of (20) with respect to and are given by

=−

− −

Z

∙(1 + )−

∗+

∗

¸∗d ()

+

−

Z

∙(1− )−

∗

¸∗

d () (21)

=−

+

−

Z

∙(1− )−

∗

¸∗

d () (22)

Using equations (4) and (5), these expressions reduce to

=−

− +

+

1

Ω( ) = 0 (23)

=−

+

+

Z

(1− )

∗

∗

d () (24)

where we define

Ω( ) ≡Z

£(1− )(

∗ ) + (1− )(

∗ )¤d () 0

to simplify notation. The elasticities of the fat tax on the harm of fat good consumption

and on the benefit of the healthy good consumption are defined as

≡ −

∗

∗

and ≡

∗

∗

Expressions (23) and (24) represent the tradeoffs which determine the optimal levels

of and . The first term of (23) reflects the marginal benefits of a higher fat tax

revenue via the induced reduction in health care premiums. The second term measures

the direct (average) utility loss associated with the price increase brought about by the

fat tax. Similarly, the third term represents the (direct) utility gain associated with

the induced subsidy on the healthy good. The last (Pigouvian) term represents the

correction for the individuals’ misperception of the health impact of the fat and healthy

good. Setting the derivative equal to zero and rearranging yields

= +Ω( )

h1− (1− )

³+ (1− ) 1

1+

´i 0 (25)

10

where ≡ − 0. In words, the utilitarian level of the fat tax is positive. This

is true even when there is no misperception of the health effects so that Ω( ) = 0.

In that case, any given individual still neglects the impact of his fat and healthy good

consumption on and this calls for a corrective tax.

Turning to (24), the first term measures the impact of on health care costs and

thus on the capitation fee. Its sign is ambiguous; see equation (17). The second and

third term are both negative. An increase in lowers which has a negative direct effect

on utility and also lowers the corrective (Pigouvian) effect of . When increases with

it is clear that increasing has only negative effects: it decreases and increases .

If instead decreases with , the solution trades off the negative effect of the decrease

in against the positive effect of a decrease in .

For the rest, equation (24) is not very informative about the sign of . This issue can,

however, be addressed in a simple way by taking a detour via the Pigouvian solution

described in Appendix A.1. Recall that the utilitarian objective yields precisely this

solution so that = and = . It then follows from equation (13) that 0

when ( ) ( ) and negative otherwise. In words, is positive if the proceeds

from the Pigouvian fat tax exceed the budgetary cost of the Pigouvian subsidy on the

healthy good.

3.2 Rawlsian welfare function

A Rawlsian social planner maximizes utility as given by (19) of the worst-off individual

in society, namely the one with the lowest income . The optimization problem then

amounts to

max

( ) = ( ) (26)

Proceeding as in the previous section, we use equations (4) and (5) to simplify the FOCs

of (26) with respect to and . They can be written as

=−

− ∗ +

∗ +

1

Ω( ) (27)

=−

+

∗ + (1− )

∗

∗

(28)

where

Ω( ) ≡ (1− )(∗) + (1− )(

∗) 0

11

The interpretations of equations (27) and (28) are similar to those of equations (23) and

(24), except that the direct utility and Pigouvian effects are now evaluated for the lowest

income individual rather than averaged over the entire population. Observe that unlike

in the utilitarian case we no longer get “straight” Pigouvian tax and subsidy rates. This

is because we are in a second-best setting with restricted instruments. Consequently, the

optimal policy is not longer determined solely by efficiency considerations; redistribution

now also matters.

Setting (27) equal to zero and rearranging gives the following expression for the

Rawlsian fat tax ,

= +Ω( )

h∗

− (1− )

³+ (1− ) 1

1+

∗

´i (29)

Since ∗ , we thus also get a positive fat tax in the Rawlsian case as long as 0.

When 0, on the other hand a negative solution to (29) cannot be ruled out, which

given our non negativity restriction on taxes and subsidies implies a corner solution

at = 0. An alternative, and more intuitive perspective on this issue is provided

by equation (A.8) in Appendix A.2. It shows that the regressive effect of the fat tax,

as measured by the term (∗ − ), may outweigh the Pigouvian terms and thus yield

= 0.

Intuitively, since the fat tax is regressive, one might expect the Rawlsian level to

be lower than the utilitarian one. However, the lowest income individual suffers most

from misperception of the adverse health effects of fat consumption. Consequently, the

Pigouvian term linked to misperception will be higher in the Rawlsian case. To show

this formally, the comparison of (25) and (29) is not very informative because the level

of will differ. We can more easily compare (A.4) and (A.8), in Appendices A.1 and

A.2. The denominator of these expressions and the first term in the numerator are the

same.9 Consequently, we have = if

(1− )0(∗)

µ−

∗

¶−Z

∙(1− )

0(∗ )µ−

∗

¶¸d () (∗ − ) (30)

In words, the Rawlsian case leads to a lower fat tax when the regressive effect, measured

by the RHS of (30), is stronger than the Pigouvian tax differential associated with the

misperception, measured by the LHS of (30). Recall that misperception increases with

9They are evaluated at different levels of but this is of no relevance. To establish the comparison

we can for instance evaluate the FOC with respect to at .

12

income. Consequently, individual ’s misperception will, roughly speaking, be larger

than the average degree of misperception and this calls for a larger tax in the Rawlsian

case.

The levels of the subsidy on the healthy good and can be compared along the

same lines, yielding again a conflict between correction for misperception and regressive

bias. This discussion also shows that even if we assume for instance and

; the comparison between and remains ambiguous. This follows from a

simple inspection of equation (13).

4 The political economy of fat taxes

We now assume that the level of the fat tax is chosen by majority voting, for a given

budgetary rule. Now, the variable which was of no direct relevance in the normative

case will be of crucial importance. With a continuum of individuals each voting agent has

zero mass implying that an individual vote cannot change the outcome of the election.

Individuals vote for the level of the fat tax that maximizes their perceived indirect utility

function as given by (18):10

max

b( ) ≥ 0

The FOC of a type- agent with respect to is given by

b

=−

− ∗ −

∙(1 + )−

∗+

∗

¸∗

+

∗ −

∙(1− )−

∗

¸∗

≤ 0

Using the FOCs of the individual choices over and , equations (4) and (5), this

equation reduces to

−− ∗ +

∗ ≤ 0 (31)

The first term represents the marginal benefit of higher fat taxes which is given by the

reduction in the capitation fee. The second term reflects the individual’s direct costs

of fat taxes. These are higher for lower income individuals since they consume more

of the fat good. The last term represents the benefit of a higher fat tax when part of

10 Individuals vote in accordance with their perceived preferences which is in contrast to Haavio and

Katakorpi (2011). In their framework individuals consume according to their perceived preferences, but

vote according to their true preferences. The voting on the fat tax is thus used as a self-commitment

device to overcome their self-control problems in the fat good consumption. See the Conclusion for

further discussion.

13

its revenue is employed to subsidize the healthy good consumption. Since low income

individuals consume less of the healthy good they also profit less from its subsidization.

For preferences to be single peaked we assume that the second-order condition is

always non-positive, i.e. ≤ 0.11 To determine the median voter, we need to knowhow the most preferred fat tax rate changes with income. Differentiating (31) we get

= −−

+

0 (32)

That is, the most preferred fat tax rate is increasing in an individual’s income. This is

intuitive: higher income individuals consume less fat goods and are thus affected less

by their taxation. In addition, high income agents profit more if some fat tax revenue

is employed to subsidize healthy good consumption. Since is a monotonic increasing

function of , the median voter is simply the individual with median income, . With

single-peaked preferences, the equilibrium fat tax corresponds to the most preferred

level of the median individual and is implicitly determined by

−− ∗ +

∗ = 0 (33)

where ∗ and ∗ are the fat and healthy good consumption of the median voter.

How does the fat tax in the political equilibrium compare to the socially optimal

level? For a utilitarian objective we have to evaluate the FOC of the social planner,

equation (23) at , i.e.

W

¯=

= (∗ − )| z 0

+(− ∗)

| z 0

+1

Ω( )| z 0

0 (34)

Assuming that the income distribution satisfies the standard property that , we

have ∗ and ∗ if is a concave function while is a convex function of .12

11Specifically, the second order condition is given by

≡ 2

2= −2

2− ∗

+

2

2∗ +

∗

≤ 0 ∀

12To see this observe that the concavity of , the property that decreases with and imply

that

≡

( ()) () ≤ ( ()) ≤ ( ()) ≡ ∗

Similarly, the convexity of and the property that is increasing in implies

≡

( ()) () ≥ ( ()) ( ()) ≡ ∗

14

Under these assumptions, the fat tax in the political equilibrium is always lower than

the social optimal fat tax, . Intuitively, this result can be understood as follows.

First, the median voter suffers more from taxation of fat goods since his consumption

is above average consumption. Additionally, he profits less from the subsidization of

healthy goods since his healthy good consumption is below average. Third, the median

voter does not take into consideration the sum of all (negative) long-run effects of fat

good consumption on society, nor the additional benefits of increased healthy good

consumption when a share of the fat tax revenue is earmarked to subsidize healthy

goods.

Turning to the Rawlsian case, evaluating the FOC for the maximization of welfare,

equation (27), at yields

W

¯=

= (∗ − ∗)| z 0

+(∗ − ∗)

| z 0

+1

Ω( )| z 0

(35)

Since ∗ ∗ , ∗ ∗, and 0, the first two terms on the RHS are negative,

whereas Ω( ) 0. Consequently, the sign of (35) is ambiguous and it is not clear

how the political fat tax compares to the socially optimal level. Unlike in the utilitarian

case, the equilibrium tax rate can be smaller than the social optimal fat tax.

Intuitively, this ambiguity arises for two reasons. First, the worst-off individual

consumes more of the fat good than the median voter and thus also suffers more from

its taxation. Second, the worst-off individual has the lowest healthy good consumption

and profits the least from its subsidization. At the same time, however, the worst-off

individual has the highest misperception with respect to the health effects of the fat and

healthy good consumption. He is thus the one who profits most from the positive health

effects that come along with a lower consumption of fat goods due to their taxation.

And this effect may drag the Rawlsian social optimal fat tax above the political fat tax.

So far, we have studied the voting equilibrium for a given level of . To study

the constitutional stage we need to know how the fat tax relates to the budgetary

rule. Earmarking fat tax revenue to the health care scheme or to the healthy good

consumption may then open up the possibility to draw the political fat tax closer to its

social optimum. The derivative of wrt is

= −−

2

+ 2

∗ +

∗

15

Rearranging yields

=

+ (1− )h∗

1(1+)

− 1 + (1−)[(1+)]

2

∗

i

= + (1− )

h∗

11+

³1 +

1+

´− 1i

(36)

Assume that 0. Roughly speaking this means that healthy consumption (or

physical activity) is more beneficial (at the margin) for individuals who consume more

fat. In other words, a heavy consumption of fat goods makes the compensation by an

otherwise healthy diet more effective. Then, a sufficient condition for (36) to be positive

is that1 + +

(1 + )2

∗

For isoelastic demand, i.e. = , this condition is satisfied and we have

0, a property we shall assume for the remainder of the paper.

5 Earmarking of fat tax revenue

So far, we have studied the voting equilibrium fat tax for a given level of . Together

with the budget constraint this also determines , the subsidy on the healthy good. We

have shown that the voting equilibrium tax is always lower than the utilitarian solution,

but that it may be lower or higher than the Rawlsian level. Furthermore, this tax

increases with . We now turn to the constitutional stage at which the earmarking rule

is determined by a welfare maximizing (utilitarian or Rawlsian) government anticipating

the induced voting equilibrium. We study if and how the earmarking rule can be used to

bring about the “best” possible political equilibrium. Intuitively, one would expect that

a utilitarian government sets a higher than otherwise optimal level of in order to boost

the political support for the fat tax.13 However, we show that the determination of the

earmarking rule is more complex than this first intuition would suggest. Specifically,

the setting of the earmarking rule may involve a conflict between the fat tax and the

healthy good subsidy.

13This intuition is in line with Casamatta et al. (2000) who deal with a problem of pension design. The

generosity is determined by voting, but the degree to which the system redistributes within generations

(the Bismarckian factor) is set at the constitutional stage. They show that it may be desirable to make

the system more Bismarckian (less redistributive) in order to boost political support.

16

5.1 Utilitarian objective

With a utilitarian objective, the constitutional problem consists in maximizing the sum

of all utilities, evaluated at the induced voting equilibrium with respect to . Formally,

the solution of is defined as

= arg max

½ () =

Z

(() )d () s.t. ≤ 1¾

Differentiating () with respect to , we have

d

d=

Z

d () +

Z

d () (37)

Since the political fat tax is too low for all possible ’s and with increasing in one

might expect that the level of set at the constitutional stage would be higher than

. In other words, we devote a higher proportion of tax proceeds to the financing of

health care (the reduction of ) than otherwise optimal. Put differently, a smaller than

optimal share would be devoted to the subsidization of the healthy good. To verify this

conjecture we have to evaluate (37) at . We know from our earlier results that the first

term on the RHS is positive and it represents the intuitive argument we just described.

Boosting will bring the supported by the median voter closer to its optimal level. In

other words, is used to boost political support for the fat tax. However, this intuition

has to be qualified because the second term on the RHS is negative. To see this, observe

that from equation (A.3) in the Appendix it follows thatZ

d () =

Z

∙−− + (1− )

0()¸

d ()

(38)

Now, we know from (16) that 0. The remainder of the RHS of (38) is simply

dWd, where W is utilitarian welfare as defined by (A.1). Now with = and

= we must have from (11) that , which from the second order condition

implies that dWd 0. Putting these elements together shows that the second term

on the RHS of (37) is negative at . Consequently, the entire expression does not

appear to have an unambiguous sign. Intuitively, this is because the increase in also

has a negative effect, namely that it further decreases which is already too low.

To sum up, while the voting equilibrium tax is always too low and while increasing

over and above the utilitarian level does boost political support for the fat tax and

increase its level, it is not necessarily optimal to do so. This is because the optimal policy

17

may involve a tradeoff between the (political equilibrium) fat tax and the healthy good

subsidy. Increasing the share of tax proceed devoted to health insurance may further

decrease the subsidy which is already too low.

5.2 Rawlsian objective

With a Rawlsian objective, the constitutional problem consists in maximizing the utility

of the worst-off individual with respect to . Formally, the solution of is defined as

= argmax

© = (() ) s.t. ≤ 1ª

Differentiating (), we get

d

d=

+

(39)

To study whether it is optimal to set below or above the Rawlsian level obtained in

the normative benchmark, we have to evaluate (39) at . Recall that unlike in the

utilitarian case, the voting equilibrium level () can now be smaller or larger than

. When () , the first term on the RHS of (39) is once again positive. And

an argument which is exactly similar to the one used in the previous subsection (using

equation (A.7) rather than (A.3) in the Appendix) can be used to show that the second

term is negative. Consequently, the sign of the overall expression is ambiguous. On the

other hand, when () , the first term on the RHS of (39) is negative. And using

once again the same argument for the second term it follows that it is now positive. To

sum up, while the respective signs of the two terms may be reversed, they always go in

opposite directions. Intuitively, this is essentially due to a simple accounting argument,

based on expression (11). Given = , when is too low, is also too low (so

that the second term is negative). But when is too large, is also too large and the

second term is positive (because an increase in brings about a decrease in the level of

in the voting equilibrium).

6 Conclusion

This paper has studied the determination of a fat tax and a subsidy on a healthy good

through majority voting. The voting procedure is preceded by a constitutional stage in

which a welfare maximizing (utilitarian or Rawlsian) government sets the earmarking

18

rule, which specifies the share of fat tax proceeds that is used to contribute to the funding

of the health care system, while the remaining revenue is used to finance the subsidy on

the healthy good. Both the long-run costs of fat consumption and the long-run benefits

of the healthy good are misperceived. In addition, individuals choose their consumption

levels without taking into account their impact on health care costs. Voting behavior is

based on the misperceived utility. In other words, the problem is not one of self-control

but one of inadequate information.

We show that the equilibrium fat tax is always lower than at the utilitarian bench-

mark, while it may be lower or higher than the Rawlsian level. Furthermore, the equi-

librium tax increases with , the share of tax proceeds devoted to health insurance.

Roughly speaking this is the case because increasing makes the overall tax policy

less regressive. Indeed, a larger subsidy on the healthy good makes the policy more

regressive.

In the utilitarian case we show that even though the earmarking rule can be used

to boost the political support for the fat tax, it is not necessarily optimal to do so.

This is because there is a conflict between the fat tax and the healthy good subsidy.

The latter is also too low at the voting equilibrium and may be further decreased as

increases. To sum up, the level of set at the constitutional stage may be lower as well

as higher than the utilitarian (full control) benchmark. Similar effects are at work in

the Rawlsian case with the added complexity that the voting equilibrium tax level may

then actually exceed the optimal level.

To sum up, earmarking is an effective tool to influence the voting equilibrium

tax/subsidy policy and bring it closer to the optimum. However, the determination

of the optimal earmarking rule is more complex then on would have expected at first.

It is not simply used to boost political support for the fat tax. Instead, it may involve a

tradeoff between the (political equilibrium) fat tax and the healthy good subsidy. Con-

sequently, the optimal level of may actually be lower than the utilitarian benchmark

in order to ensure an more adequate level of the healthy good subsidy.

We have assumed that individuals vote according to their misperceived preferences.

Haavio and Kotakorpi (2011) investigate a problem of self-control over sin goods and

consequently assume that the median voter votes accordingly to his long-term utility

function. In our setup this translates into assuming that individuals vote based on

19

preferences with = = 1. In this case the median voter chooses the fat tax while

being aware of his own misperception of the utility costs and benefits of fat and healthy

goods.14 In other words he uses the tax policy as a commitment devise to mitigate

his “self-control” problem. Consequently the first of the two externalities described

in Section 3 is accounted for, at least in part.15 The second, health insurance cost,

externality remains however relevant. Overall we can thus expect to be further

away from the utilitarian choice when individuals vote according to their misperceived

utility function. On that account we believe that understanding if individuals are af-

fected by misperception–and vote accordingly to their misperceived preferences–or

by temptation–and vote accordingly to the true effects–has important policy implic-

ations.

14The analogous of (33) would be

−

−

∗(1− )

∗

∗

+

∗ − (1− )

∗

∗

= 0 (40)

It comes out that the third term in (34) would be partially corrected by the median voter’s misperception

terms.15To be more precise the commitment benefit is mitigated by the tax revenue effect which arises

because the median voter does not get a full refund of his tax payments.

20

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A Appendix

A.1 Pigouvian taxes and subsidies

To obtain the Pigouvian levels and , we solve

max

W( ) =Z

[ − − (1 + )∗ − (1− )∗ + (∗ ) + (∗ )− (∗ )]d ()

s.t. + () = ( ) + ()

substituting from the budget constraint and adding and subtracting (∗ )− (

∗ ),

W can be rewritten as

W =

Z

[ − ( )− () + ()− (1 + )∗ − (1− )∗ + (∗ ) + (∗ )

−(∗ ) + (1− )(∗ )− (1− )(

∗ )]d () (A.1)

Differentiating and rearranging by making use of the envelope theorem yieldsZ

∙−

∗+ − (1− )

0(∗ )¸∗d () =0 (A.2)Z

∙−

∗− + (1− )

0(∗ )¸∗d () =0 (A.3)

rearranging and “solving” yields

=

∗

+(1−)0(∗ )

−∗

d ()

−

0 (A.4)

=

−∗∗

+(1−)0(∗ )

∗d ()

0 (A.5)

A.2 Rawlsian taxes and subsidies

To obtain the Rawlsian solution by using taxes and subsidies as decision variables we

determine and , by

max

W( ) = [ − − (1 + )∗ − (1− )∗ + (∗) + (∗)− (∗)]

s.t. + () = ( ) + ()

24

substituting from the budget constraint and adding and subtracting (∗)− (

∗),

W can be rewritten as

W =∗ − ( )− () + ()− (1 + )∗ − (1− )∗ + (∗) + (∗)− (

∗)

+ (1− )(∗)− (1− )(

∗)

Differentiating and rearranging, by making use of the envelope theorem, yieldsZ

∙−

∗+

¸∗d () + (1− )

0(∗) + (− ∗) = 0 (A.6)Z

∙−∗−

¸∗d () + (1− )

0(∗) + (∗ − ) = 0 (A.7)

rearranging and “solving” yields

=

R

∗

³−∗

´d () + (1− )

0(∗)³−∗

´− (∗ − )

−

(A.8)

=

R −

∗∗d () + (1− )

0(∗)∗

− (− ∗)

(A.9)

The denominators of both of these expressions are positive. The first two terms in the

numerator of (A.8) are positive Pigouvian terms which correct for the two externalities

that are present in our setting. The first measures the externality via health care costs

and the second the correction for misperception. The third term, however, is negative

and measures the regressive effect of the Pigouvian tax. The poorest individual pays

∗ but his refund via the reduction in is only . The numerator of (A.9) has a

similar interpretation with two Pigouvian and a third redistributive term, which once

again is negative.

25